[Paper Review] Approach and separation of quantum vortices with balanced cores

C. Rorai, Jack W. D. Skipper+2 more|arXiv (Cornell University)|Oct 6, 2014

Quantum, superfluid, helium dynamics11 references3 citations

TL;DR

This study uses high-precision numerical simulations of the 3D Gross-Pitaevskii equation to investigate quantum vortex reconnection, introducing improved vortex core tracking and initial condition control to reveal distinct scaling laws for anti-parallel and orthogonal vortex configurations. The key finding is that orthogonal vortices exhibit anomalous pre- and post-reconnection scaling of δ ∼ |tₜ − t|¹ᐟ³ and δ ∼ |t − tₜ|²ᐟ³, respectively, differing from the dimensional prediction δ ∼ |tₜ − t|¹ᐟ², with these differences linked to early, persistent Frenet-Serret frame misalignments.

ABSTRACT

Using two innovations, smooth, but distinctly different, scaling laws for the numerical reconnection of pairs of initially orthogonal and anti-parallel quantum vortices are obtained using the three-dimensional Gross-Pitaevskii equations, the simplest mean-field non-linear Schrödinger equation for a quantum fluid. The first innovation suppresses temporal fluctuations by using an initial density profile that is slightly below the usual two-dimensional steady-state Padé approximate profiles. The second innovation is to find the trajectories of the quantum vortices from a pseudo-vorticity constructed on the three-dimensional grid from the gradients of the wave function. These trajectories then allow one to calculate the Frenet-Serret frames and the curvature of the vortex lines. For the anti-parallel case, the scaling laws just before and after reconnection obey the dimensional $δ\sim|t_r-t|^{1/2}$ prediction with temporal symmetry about the reconnection time $t_r$ and physical space symmetry about the $x_r$, the mid-point between the vortices, with extensions of the vortex lines formng the edges of an equilateral pyramid. For all of the orthogonal cases, before reconnection $δ_{in}\sim(t-t_r)^{1/3}$ and after reconnection $δ_{out}\sim(t-t_r)^{2/3}$, which are respectively slower and faster than the dimensional prediction. In these cases, the reconnection takes place in a plane defined by the directions of the curvature and vorticity. To define the structure further, lines are drawn that connect the four arms that extend from the reconnection plane, four angles $θ_i$ between these arms are found, then summed, giving $\sumθ_i>360^\circ$. This implies that the overall structure is convex or hyperbolic, as opposed to the acute angles of the anti-parallel pyramid.

Motivation & Objective

• To resolve the lack of direct microscopic insight into quantum vortex reconnection dynamics in quantum fluids.
• To address the challenge of accurately tracking vortex core positions and minimizing numerical fluctuations in simulations of vortex reconnection.
• To investigate whether different initial configurations—anti-parallel versus orthogonal vortices—lead to distinct scaling laws for vortex separation during reconnection.
• To explore the geometric and kinematic origins of anomalous scaling laws observed in orthogonal vortex reconnection.
• To link the observed scaling behaviors to the evolution of Frenet-Serret frame alignments and curvature dynamics.

Proposed method

• Numerical integration of the three-dimensional Gross-Pitaevskii equation using a split-step spectral method to simulate quantum fluid dynamics.
• Implementation of a pseudo-vorticity field constructed from wave function gradients to accurately locate vortex core positions on a 3D grid.
• Use of a tailored initial condition with balanced density profiles around vortex cores to suppress spurious temporal density fluctuations and stabilize scaling behavior.
• Computation of vortex line trajectories, curvature, and Frenet-Serret frames to analyze local geometric and kinematic properties during reconnection.
• Definition of reconnection plane via average curvature and vorticity directions at closest approach, and analysis of angles between extending vortex arms.
• Comparison of scaling laws between anti-parallel and orthogonal initial configurations, focusing on temporal evolution of minimum separation δ(t).

Experimental results

Research questions

• RQ1 Do different initial configurations of quantum vortices—specifically anti-parallel versus orthogonal—lead to distinct scaling laws for the minimum separation distance during reconnection?
• RQ2 Why do orthogonal vortex reconnections exhibit scaling laws that deviate from the dimensional prediction δ ∼ |tₜ − t|¹ᐟ²?
• RQ3 How do the geometric alignments of the Frenet-Serret frames influence the observed scaling behavior in vortex reconnection events?
• RQ4 To what extent do the anomalous scaling laws persist across different initial separations, and could they apply to macroscopic-scale vortex systems?
• RQ5 Can the observed scaling differences be linked to the global alignment structure of the vortex lines, particularly the sum of angles formed by the four arms extending from the reconnection plane?

Key findings

• For initially anti-parallel vortices, the separation distance δ(t) follows δ ∼ |tₜ − t|¹ᐟ² both before and after reconnection, showing temporal symmetry about tₜ and physical space symmetry about the midpoint.
• For initially orthogonal vortices, the pre-reconnection scaling is δ ∼ |tₜ − t|¹ᐟ³ (slower than dimensional), and post-reconnection scaling is δ ∼ |t − tₜ|²ᐟ³ (faster than dimensional), indicating anomalous behavior.
• The anomalous scaling laws for orthogonal vortices are not transient but persist from early times and are tied to a persistent hyperbolic alignment of the Frenet-Serret frames, with the sum of angles between extending vortex arms exceeding 360°.
• The reconnection plane for orthogonal vortices is defined by the average directions of curvature and vorticity at closest approach, and the four arms extending from it form a convex, hyperbolic structure.
• The observed scaling laws are robust across different initial separations and are not artifacts of numerical noise, as confirmed by the use of a core-balanced initial condition suppressing unwanted fluctuations.
• The results suggest that the anomalous scaling for orthogonal vortices may extend to macroscopic scales, as the geometric alignment persists throughout the reconnection process and is independent of initial separation distance.